Tauberian Theory of Wave Fronts in Linear Hereditary Elasticity

This book provides a rigorous mathematical approach to linear hereditary problems in wave propagation theory and demonstrates the efficiency of mathematical theorems in hereditary mechanics. It uses real and complex Tauberian techniques for the Laplace transform to classify near-front asymptotics of solutions and derive the behavior of these solutions.

Format: Paperback / softback
Length: 136 pages
Publication date: 23 October 2021
Publisher: Springer Verlag, Singapore

The aim of this comprehensive book is to establish a rigorous mathematical framework for addressing linear hereditary problems in wave propagation theory, while also showcasing the remarkable effectiveness of mathematical theorems in the realm of hereditary mechanics. Through the application of both real and complex Tauberian techniques for the Laplace transform, a comprehensive classification of near-front asymptotics of solutions to the considered equations is provided, contingent upon the singularity characteristics of the memory function. The book proceeds to derive a detailed description of the behavior of these solutions, emphasizing the significance of nonlinear Laplace transforms in linear hereditary elasticity. This invaluable resource is of immense importance to researchers engaged in the fields of mathematical physics and related disciplines.

Introduction:
The study of wave propagation in various physical contexts has been an essential aspect of mathematics and physics for centuries. In particular, the analysis of waves propagating through elastic media has gained significant attention due to its wide range of applications, including engineering, geology, and biology. Linear hereditary problems, which arise from the study of wave propagation in elastic media, play a crucial role in understanding the behavior of these systems. These problems involve the propagation of waves through a medium that exhibits both linear and elastic properties, and they are governed by a set of partial differential equations (PDEs).
The Objective of the Book:
The primary objective of this book is to construct a rigorous mathematical approach to linear hereditary problems of wave propagation theory and demonstrate the efficiency of mathematical theorems in hereditary mechanics. By using both real and complex Tauberian techniques for the Laplace transform, a classification of near-front asymptotics of solutions to considered equations is given—depending on the singularity character of the memory function. The book goes on to derive the description of the behavior of these solutions and demonstrates the importance of nonlinear Laplace transform in linear hereditary elasticity.
Classification of Near-Front Asymptotics:
The book begins by discussing the concept of near-front asymptotics, which refers to the behavior of solutions to PDEs as they approach a certain point or boundary. Using both real and complex Tauberian techniques, the author provides a comprehensive classification of near-front asymptotics for solutions to linear hereditary problems. This classification is based on the singularity character of the memory function, which determines the type of behavior exhibited by the solutions.
Derivation of Behavior of Solutions:
Once the classification of near-front asymptotics is established, the book goes on to derive a detailed description of the behavior of these solutions. This involves analyzing the PDEs and using various mathematical techniques, such as integral transforms, to obtain the desired solutions. The author emphasizes the importance of understanding the behavior of these solutions in order to predict the behavior of elastic media under different conditions.
Importance of Nonlinear Laplace Transform:
One of the key insights of the book is the importance of nonlinear Laplace transform in linear hereditary elasticity. The author demonstrates how the nonlinear Laplace transform can be used to analyze the behavior of elastic media and to obtain solutions to complex problems. This approach is particularly useful in situations where traditional methods of analysis are not applicable or insufficient.
Conclusion:
In conclusion, this book is of undeniable value to researchers working in areas of mathematical physics and related fields. By providing a rigorous mathematical framework for addressing linear hereditary problems in wave propagation theory, the author showcases the remarkable effectiveness of mathematical theorems in the realm of hereditary mechanics. The book's comprehensive classification of near-front asymptotics, derivation of behavior of solutions, and emphasis on nonlinear Laplace transform make it an essential resource for anyone interested in the study of elastic media and wave propagation.

Weight: 238g
Dimension: 235 x 155 (mm)
ISBN-13: 9789811585807
Edition number: 1st ed. 2020